Division of the perfect fifth (3/2) into n equal parts

Division of the 3:2 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of equivalence is still in its infancy. The utility of 3:2 as a base though, is apparent by being one of the strongest consonances after the octave. Many, if not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Perhaps the first to divide the perfect fifth was Wendy Carlos ( http://www.wendycarlos.com/resources/pitch.html). Carlo Serafini has also made much use of the alpha, beta and gamma scales.

Incidentally, one way to treat 3/2 as an equivalence is the use of the 8:9:10:(12) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes six 5/4 to get to 9/8 (tempering out the comma 15625/15552. So, doing this yields 9, 11, and 20 note MOS which the Carlos scales temper equally. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it if it hasn't been named yet, but in any case here is an example of it.

Individual pages for EDFs | EDO-EDF correspondence

---

Individual pages for EDFs

4edf
5edf
6edf
7edf
8edf (88cET)
9 (Carlos Alpha)
10edf
11 (Carlos Beta)
12edf
13edf
14edf
15edf
16edf
17edf
18edf
19edf
20 (Carlos Gamma)
21edf
22edf
23edf
24edf
25edf

EDO-EDF correspondence

EDO	EDF	Comments
7edo	4edf	4edf is 7edo with 28.5 cent stretched octaves. Equivalently, 7edo is 4edf with 3/2s compressed by ~16 cents. Patent vals match through the 5 limit. Only a rough correspondence.
8edo
9edo	5edf	Very rough correspondence - patent vals disagree in the 5 limit.
10edo	6edf	Also very rough.
11edo
12edo	7edf	7edf is 12edo with 3.4 cent stretched octaves. Equivalently, 12edo is 7edf with 2.0 cent compressed 3/2s. With the exception of 11 (which falls almost exactly halfway between steps in both cases), the patent vals match through the 31 limit, so the agreement is excellent.
13edo
	8edf	Since 88cET/octacot is well known to approximate some intervals quite accurately, it would be wrong to lump this in with 14edo.
14edo
15edo
	9edf	The Carlos alpha scale is neither 15edo nor 16edo.
16edo
17edo	10edf	10edf is 17edo with 6.6 cent compressed octaves. Patent vals match through the 13 limit, with the exception of 5 (as expected).
18edo
19edo	11edf	11edf is 19edo with 12.5 cent stretched octaves. Patent vals match through the 7 limit. If you don't think Carlos beta is accurately represented by 19edo then ignore this correspondence.
20edo
	12edf	12edf entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\41 of an octave.
21edo
22edo
	13edf	Perhaps surprisingly, this is not very similar to 22edo. Patent vals differ in the 5 limit.
23edo
24edo	14edf	14edf is 24edo with 3.4 cent stretched octaves. Patent vals agree through the 19 limit.
25edo
26edo	15edf	Fairly rough correspondence. 15edf is 26edo with ~17 cent stretched octaves. Patent vals agree through the 5 limit, but not through the 7 limit.
27edo
	16edf
28edo
29edo	17edf	17edf is 29edo with 2.5 cent compressed octaves. Patent vals disagree in the 7 limit.
30edo
	18edf	Perhaps surprisingly, this is not very similar to 31edo. Patent vals differ in the 5 limit.
31edo
32edo
	19edf
33edo
34edo	20edf	20edf is 34edo with 6.6 cent compressed octaves. Patent vals match through the 5 limit, but not the 7 limit. If you don't think Carlos gamma is accurately represented by 34edo then ignore this correspondence.
35edo
36edo	21edf
37edo
38edo	22edf
39edo	23edf
40edo
41edo	24edf
42edo
43edo	25edf